Barney L. Bales,* Michelle Meyer, Steve Smith, and Miroslav Peric

Transcription

1 4930 J. Phys. Chem. A 2009, 113, EPR Line Shifts an Line Shape Changes ue to Spin Exchange of Nitroxie Free Raicals in Liquis: 6. Separating Line Broaening ue to Spin Exchange an Dipolar Interactions Barney L. Bales,* Michelle Meyer, Steve Smith, an Miroslav Peric Department of Physics an Astronomy an The Center for Supramolecular Stuies, California State UniVersity at Northrige; Northrige, California ReceiVe: October 23, 2008; ReVise Manuscript ReceiVe: January 12, 2009 EPR spectra of pereuterate 2,2,6,6-tetramethyl-4-oxopiperiine-1-oxyl (PDT) are stuie as functions of molar concentration, c, an temperature, T, in water an 70 wt % glycerol in water. The increase of the intrinsic line with average over the three hyperfine lines, B tot, varies linearly with c with zero intercept in both solvents at all temperatures; therefore B tot /c is inepenent of c. The spin exchange inuce ersion, from which the spin exchange frequency, ω e, may be compute, increases linearly with B tot, passing through the origin in water an in 70% glycerol at high temperatures; however, at low temperatures, where ipolar interactions broaen the spectra, linearity oes not prevail until B tot > 1 G ue to a contribution of ipolar interactions to the ersion. The broaening constant ue to spin exchange, B e /c, is foun from the slope of the linear region, permitting a computation of the ipolar constant, B ip /c ) B tot /c - B e /c. Thus, the separation of concentration broaening into spin exchange an ipolar contributions is effecte without having to appeal to some suppose temperature epenence of the two interactions. The fractional broaening by spin exchange, Ω(T), is near unity at high temperatures in both solvents, ecreasing to zero in 70% glycerol at 273 K. Ω(T) is a continuous function of the inverse rotational correlation time of PDT but is iscontinuous as a function of T/η where η is the shear viscosity. Ω(T) ) 0.5, where spin exchange an ipolar interactions contribute equally to the line with occurs at T/η ) 20 ( 1 K/cP in 70% glycerol. Hyroynamic preictions of B e /c via the Stokes-Einstein (SE) equation are remarkably accurate in 70% glycerol comparable with the results in a series of alkanes. In water, B e /c is linear with T/η with zero intercept as require by the SE; however, with slope a factor of 0.73 smaller. B ip /c is reasonably preicte by the SE only at very small values of η/t very quickly following an approximately logarithmic epenence rather that the linear preiction. Values of B ip /c approach a plateau above η/t ) 0.20 cp/k that is about one-half the soli state limit. Line shifts ue to spin exchange are not yet useful to euce values of Ω(T) ue to a lack of knowlege of the time between re-encounters; however, they may be use to verify the values etermine from line broaening an spin exchange inuce ersion. Some effects at low temperatures in 70% glycerol suggest that the effects of ipolar interaction are inaequately escribe by the wiely accepte theory. Introuction This is a continuation of a series of articles (parts 1-5) 1-5 whose ultimate goal is to measure the translational iffusion of nitroxie free raicals (nitroxies) in compartmentalize systems, such as micelles. We began the work 18 years ago 6 thinking that the theory was complete: 7 spin exchange broaene an shifte the resonance lines when ω e /γ, A 0 an narrowe them in the opposite extreme, where ω e is the spin exchange frequency, A 0 the hyperfine coupling constant in the absence of spin exchange, an γ the gyromagnetic ratio of the electron. It was thought that ω e coul be erive from any one of these three effects, 7 an we thought that we coul use a combination of the effects to euce collision frequencies an statistical istributions of nitroxies in compartmentalize systems. However, the preicte line shifts turne out not to agree with experiment 2,3,8-11 an the narrowing is not practical to use in most interesting cases, 7 leaving line broaening as the only tool. It was known 7 that the line shape eparte from Lorentzian as ω e increase, so we turne our attention to that aspect of the * To whom corresponence shoul be aresse. barney.bales@ csun.eu. Webpage: vcphy00s/bbvita.html. problem an learne that the line shape coul be expresse as a sum of two components, an absorption an a spin exchange inuce ersion. 1 In part 2, 2 we showe that only two components were necessary to reprouce the EPR spectrum to very high values of ω e where ω e /γa 0 was near unity an the lines coalesce into a single line, an in part 3 we propose that the line shift iscrepancy was ue to re-encounter collisions. 3 Part 4 4 provie a test of the line shift proposal, an part 5 5 stuie re-encounters in a series of alkanes. In short, at high values of T/η, where T is the absolute temperature an η is the shear viscosity, we finally think that the theory is complete, although more work is neee to fully establish that the reencounter mechanism is responsible for the line shift iscrepancy. 12 We refer the reaer to parts 1-5 an the references therein for extensive iscussion. 1-5 A major remaining problem arising as T/η ecreases - separation of ipolar an spin exchange interactions - is the subject of the present work. This problem often occurs in the most interesting liquis an complex fluis because biologically important systems are often viscous enough to require such a separation. 13 Applications of spin-spin interactions between nitroxies in biological an moel membranes are numerous /jp CCC: $ American Chemical Society Publishe on Web 04/01/2009

2 EPR Line Shifts an Line Shape Changes J. Phys. Chem. A, Vol. 113, No. 17, TABLE 1: Results of Fitting Spectra Generate by the Rigorous Theory to Eq 1 for 14 N Nitroxies (Dispersion an Line Shifts) 4ω e /(3 3γA 0 ) a (V ex (()/ (()) b b c b b A abs /A 0 A abs /A 0 B e (0)/A 0 B e (()/A 0 B e (0) /A 0 B e (()/B e (0) I(() b I(0) b I(()/I(0) a Input values. Compare with column 5. b Fitting simulate spectra to eq 1. c Compute from the perturbation values A abs /A 0 ) 1 - ( 9 / 32 )(B e /A 0 ) 2. Both spin exchange an ipolar interactions broaen EPR lines linearly with concentration. 13 In the past, most of the efforts to separate the interactions ha to rely on a suppose temperature epenence of the ipolar interaction often formulate from the Stokes-Einstein equation. 13 Unfortunately, a paper publishe in 1979 by Berner an Kivelson 13 showe that the line broaening metho was not very promising because the line with i not show a hyroynamic epenence at low values of T/η. At first, our strategy to separate ipolar an spin exchange interactions was to measure spin exchange inuce ersion to euce ω e, use this to compute the broaening ue to spin exchange, an subtract from the total broaening to get the broaening ue to ipolar interaction. Then we planne to use the line shifts to get an inepenent value of ω e to ouble check the proceure; however, this went awry as mentione in the first paragraph. Therefore, we are left with just enough information to be able to separate the effects ue to spin exchange an ipolar interactions at a single temperature not having to make any assumptions about the moel of iffusion. Fortunately, we are able to utilize the line shifts to verify the results. Theory. Heisenberg Spin Exchange. The theory has been presente in etail for EPR spectra ue to 14 N nitroxies unergoing spin exchange in low-viscosity liquis. 1-3,7 We briefly summarize the results an refer the reaer to the literature for more etails an iscussion. 1-5 The generally accepte expression escribing the spectrum, given by Currin 15 in 1962, preicts that the lines broaen, shift towar the center of the spectrum, an eventually merge into a single line as ω e increases. We call Currin s 15 expression, erive by many workers, 1 the rigorous theory. For a summary, see ref 1 an the appenix of the same reference 1 for explicit expressions of the rigorous theory written for a first-erivative magnetic fiel-swept spectrum. The rigorous theory is in agreement with experiment not only for two- an three- but also for five-line spectra in every etail except one. 2-4 The line shifts are larger than preicte in every case 2-4 except one, aqueous solutions of Fremy s salt. 1 We showe that the line shapes in the first-erivative fiel-swept form preicte by the rigorous theory were reprouce to high precision by a simple analytical expression vali all the way up through line coalescence as follows 2-4 Y(H, ω e ) ) (M I )S MI (H) + V ex (M I )D MI (H) (1) M I where, H is the swept magnetic fiel, S MI (H) is the absorption, an D MI (H) is a spin-exchange inuce ersion. The sum in eq 1 is over the nuclear quantum number M I )+1, 0, an -1, labeling the low-, central-, an high-fiel lines, respectively. Eq 1 is ientical to eq 1 of ref 2 except that the absorption of Lorentzian shape has been replace by S MI (H), which is a Gaussian-Lorentzian sum function approximation to a Voigt shape given by eq 12 of ref 16. The absorptions S MI (H) involve mixing parameters from which the separate Lorentzian an Gaussian contributions to the linewiths may be euce an various parameters may be correcte for inhomogeneous broaening. 16 is the peak-to-peak amplitue of the absorption an M I V ex (M I ) are the maximum values of D MI (H). The expression for D MI (H) is given by eq 10 of ref 2. The creit for the form of eq 1 ought to go to Molin et al. 7 who worke out the perturbation theory preicting the secon term. Our contribution was to apply the term to nitroxies, evelop nonlinear leastsquares fitting software allowing us to measure it, an permit the parameters in the original perturbation theory to vary with ω e to preserve the simple form at high values of (ω e )/(γa 0 ). 1,2 Tables 1 an 2 give the results of fitting spectra generate by the rigorous theory to eq 1 for 14 N an 15 N nitroxies, respectively. Inhomogeneous broaening by unresolve hyperfine structure has no effect on the results in Tables 1 an 2. 2 In a conventional first-erivative presentation, such as that in Figure 1 below, where the integrate spectrum is positive, V ex (+)/ (+) )-V ex (-)/ (-) > 0, whereas for the center line V ex (0) ) 0. The ratios V ex (M I )/ (M I ) are monotonic functions of (ω e )/(γa 0 ) that are well approximate by the following:

3 4932 J. Phys. Chem. A, Vol. 113, No. 17, 2009 Bales et al. ω e )( 3 3 γa 0 4 ( V ex (() (() )[ ( V ex 2 (() TABLE 2: Results of Fitting Spectra Generate by the (() ) ] (2) Rigorous Theory to Eq 1 for 15 N Nitroxies (ω e )/( 3γA 0 ) a ((V ex (()/ (()) b b c A abs /A 0 A abs /A 0 The leaing term in eq 2 is preicte by perturbation theory 1,7 an the secon, foun by fitting the values in Tables 1 an 2, corrects 3 (ω e )/(γa 0 ) to within 0.5% for (ω e )/(γa 0 ) e For more precision or to fin higher values of (ω e )/(γa 0 ), interpolation of the values in Tables 1 an 2 may be employe. In aition to the spin exchange inuce ersion component in eq 1, an instrumental ersion component ue to an imbalance in the microwave brige often appears in the spectrum that may be correcte as follows: 4 V ex (()/ (() ) V (()/ (() - V (0)/ (0) (3) There are two inepenent values of the right-han sie of eq 2, which we abbreviate as follows: ( V ex # )( (( V ex (() (() )[ ( V ex 2 (() (() ) ] (4) To save space, we shorten the terms spin-exchange inuce ersion to exchange ersion, spin exchange to exchange, an ipolar interactions to ipolar when the meaning is clear. Dipolar. The effect of ipolar on spin relaxation in liquis at large values of T/η where motional narrowing theory is expecte to hol has been available for many years 17 an applie to electron spins at the X-ban. 13 The effect of ipolar in the static limit where T/η approaches zero is also known. 13 In both of these limits, the lines are Lorentzian in shape an broaen linearly with the molar concentration, c. In recent years, Nevzorov an Free have brige the gap between the motional narrowing regime an the rigi limit by consiering couple systems of two spins 18 an many-boy systems. 19 These later theories have not yet avance to the point of being applicable to nitroxies; however, they confirm that the lines broaen in proportion to c an are approximately Lorenztian. Assumptions. Thus, summing up the oler theories, we assume that ipolar (1) prouces broaening that is proportional to c an is inepenent of M I ; (2) prouces Lorentzian lines; (3) contributes neither to V ; nor to (4) line shifts; an (5) oes not alter the relative intensities of the three lines. The above assumptions follow from the wiely accepte theory of the equations of motion of the transverse components of magnetizations that inclue only secular terms, thereby preventing coherence transfer that is responsible for the interesting effects in spin exchange outline above. We are inebte to one of the referees who calle our attention to more recent work, 20 where the authors argue that nonsecular terms are require in the ipolar interaction. Such terms woul, in principle, also lea to ersion, line shifts, an so forth; however, with the important ifference that the signs of the ipolar nonsecular terms are opposite to those for spin exchange. The theory is not yet available for the three-line case pertinent to nitroxies; however, qualitatively, the lines are expecte to shift towar one another an merge, but not narrow as the ipolar interaction increases. 20 Dispersion terms are expecte a Input values. The broaening measure from the simulate spectra yiels values of (B e /A 0 ) that are within 10-8 of (ω e )/( 3γA 0 ). b Fitting simulate spectra to eq 1. c Compute from the perturbation values A abs /A 0 ) 1 - ( 3 / 2 )(B e /A 0 ) 2. of opposite sign to those uner spin exchange. These two manifestations of ipolar interactions woul negate assumptions 3 an 4. We o not yet know whether the secon part of assumption 1 or assumption 5 woul be negate. The arguments in ref 20 seem soun; however, we o not know of any experimental confirmation of the etails of the theory. In this work, we are unable to confirm the theory but fin intriguing evience in favor of some of its aspects, which offers some interesting prospects for future work. Separation of Exchange an Dipolar. The broaening of the M I line is compute from B tot (M I ) ) H L pp (c) MI - H L pp (0) MI (5) where H L pp (c) MI an H L pp (0) MI are the peak-to-peak intrinsic Lorentzian linewiths in the presence an absence of concentration broaening ue to exchange an ipolar, respectively.

4 EPR Line Shifts an Line Shape Changes J. Phys. Chem. A, Vol. 113, No. 17, Exchange broaens the M I )(1 lines slightly faster than the M I ) 0 line (Table 1). 3 We efine B e ) 1 / 3 [B e (+) + B e (0) + B e (-)] with B tot similarly efine. Line broaening ue to the two interactions are aitive, 13 thus Exchange broaens the lines as follows: Because both ipolar an exchange broaen the lines linearly with concentration, at a given temperature in a given solvent, the fractional broaening by exchange is expecte to be inepenent of the molar concentration, c. Near c ) 0, it has been observe that linearity breaks own so following Berner an Kivelson, 13 we take the erivative of eq 6 with respect c an employ eq 7 to write B tot ) B e + B ip (6) B e ) 4 ω e 3 3 In terms of the broaening constants B tot /c, B e /c ) ([A 0 (V ex / )] # ( /c, an B ip /c, we write from eq 9 The terms on the right-han sie of eq 10 are measure irectly an inepenently allowing us to euce B ip /c. Note that the commonly quote secon-orer spin exchange rate constant 7 may be compute as follows: K e ) (3 3γ/4)( B e /c). In the region in which Ω(T)is constant versus c, we may write In eq 11, uncertainties in c an in the temperature yiel uncertainties in Ω(T) that can be minimize by observing that Therefore, inepenent measurements of the total broaening an the ersion amplitue yiel two inepenent values of Ω(T) (. Note that assumptions 1 an 3 are employe in the erivation of eq 12. In principle, line shifts coul also be use to provie another inepenent measure of ω e although of slightly lower precision. 1 There are two shifts: one that ecreases the spacing between resonance fiels, A abs ; the other that ecreases the spacing ex γ )(A 0( V # )( (7) Ω(T) B e /B tot (8) c ( B tot ) )( c A 0( V Ω(T) ( ) ex # )( + c (B ip ) (9) c B ip ) c B tot - c B e (10) c ( B e /A 0 ) ) c ( B tot /A 0 ) Ω(T) ( )( ( c( V ex # )( (11) c ( B tot /A 0 ) ex B tot A 0( V # )( (12) between the fiels where the spectrum crosses the baseline, A obs. See Figure 1, where 2A abs is epicte as the spacing between the maxima of the ersion features; however, it is also the spacing between the absorption lines that are not shown. A abs cannot be measure irectly from the spectrum; it must be euce from a fit to eq 1. The shifts leaing to A obs are about a factor of 3 larger than those leaing to A abs. See earlier papers for explicit expressions for these spacings as preicte by perturbation theory. 1 The expressions for A abs are given in the footnotes to Tables 1 an 2. Note that the ifference between the perturbation theory preictions of A abs an the rigorous values are extremely small (compare columns 3 an 4 of Tables 1 an 2, respectively). The rigorous theory oes not inclue the effect of exchange ue to re-encounter collisions an is not in agreement with experiment except in one case, Fremy s salt. 1 The expression for A abs, moifie by inclusion of a parameter escribing the effect of re-encounters on the line shifts, κ ) ( 3/8)(1 + 2)(( γa 0 τ RE /2), where τ RE is the mean time between re-encounter collisions is given in eq 8 of ref 4, whereas the same moification of A obs is as follows: where H L pp (0) ( is the average Lorenztian line with of the lowan high-fiel lines in the unbroaene spectrum. The aitional term, -κ(b e /A 0 ), applies only in the case of strong exchange, that is when ω e is near the collision frequency. 12 We have focuse almost entirely on A abs in recent work because the effect ue to re-encounter exchange is a larger fraction of the total shift, allowing us to stuy re-encounters. 3-5 Here, our motive is ifferent: we wish to stuy A obs because the term -κ(b e /A 0 ) is quickly ominate by the final term in eq 13, masking the small temperature epenence of κ. 4 Therefore, a plot of A obs /A 0 versus B e /A 0 will yiel a universal curve for all temperatures an concentrations except near the origin. This will provie an inepenent check on the values of B e obtaine from measurements of ( (V ex / ) # (. The translational iffusional coefficient of a Brownian particle iffusing in a bulk liqui, is given in the hyroynamic limit by the Stokes-Einstein (SE) relation where a is the effective hyroynamic raius an k the Boltzmann constant. Utilizing the SE relation, Berner an Kivelson 13 showe that an A obs ) 1 - A 0 ( 9 16 H B L pp(0) ( + κ ) e - 27 A 0 32( B 2 e A 0) for low values of η/t an (13) D ) kt/6πaη (14) c B e SE ) C e T/η (15) c B ip SE ) C ip η/t (16) c B ip SE static ) C ip (17)

5 4934 J. Phys. Chem. A, Vol. 113, No. 17, 2009 Bales et al. Figure 1. EPR spectra ue to 0.1 (top trace, at a relative gain of x2) an 93.6 mm (mile trace) PDT in 70% glycerol at 343 K. The fit of the mile trace to eq 1 is inistinguishable from the spectrum; the spectrum minus the fit gives the resiual shown as the noisy baseline. The fit gives three absorption (not shown) an three ersion lines that are given in the lower trace. This is the stanar presentation of the spectrum, yieling a positive absorption spectrum upon integration. The ersion component on the central line is ue to an imbalance in the microwave brige; the component at low-fiel with a maximum value, V obs (+1) > 0, an the one at high-fiel with V obs (-1) < 0 are expecte from exchange. Exchange shifts the lines towar the center, resulting in line spacings 2A obs an 2A abs that are less than the spacing 2A 0 observe in the absence of exchange. The spectrum is ominate by exchange; the fractional broaening by spin exchange is Ω(T) ) ( Figure 2. EPR spectra ue to 0.1 (top trace, at a relative gain of 8) an 93.6 mm (mile trace) PDT in 70% glycerol at 293 K. The fit an spectrum are inistinguishable; however, a efinite resiual is evient ue to a significant ipolar interaction; the fractional broaening by spin exchange is Ω(T) ) 0.30 ( in the soli state limit as η/t f. For nitroxies, evaluating the constants in Berner an Kivelson s 13 eqs 8, 10, an 11, static C e ) cp G/M eg, C ip ) 763 eg G/cP M, an C ip ) G/M. In evaluating C e, we have assume that the istance for exchange interaction is equal to the hyroynamic iameter of the probe. Experimental PDT (lot # A559P2, pereuterate 2,2,6,6-tetramethyl-4- oxopiperiine-1-oxyl, 97 atom % ) was purchase from CDN Isotopes an use as receive. ACS reagent glycerol (99.8%) was obtaine from Sigma an use as receive. Stock solutions of 19.9 mm in water an 93.6 mm in 70 wt % aqueous glycerol were prepare in MilliQ water by weight an ilute. Unegasse samples were seale into 50 µl pipettes an place into sample tubes house in Bruker s nitrogen gas-flow temperature control ewar. The accuracy of the concentrations is estimate to be (3% an the relative concentrations (1%. The temperature, accurate to (1 K, was measure with a thermocouple place just above the active portion of the microwave cavity. During each spectrum, the temperature was stable to (0.1 K. Three EPR spectra were acquire, one after the other, with a Bruker 300 ESP X-ban spectrometer interface with Bruker s computer employing a sweep time of 41 s; time constant, 10 ms; microwave power, 5 mw; sweep with, 50.2 G; an moulation amplitue, 0.2 G. The magnetic sweep with was measure with Bruker s NMR Gaussmeter operating in the 1 mg resolution moe an was average over a ay s run. At low concentrations, the line shapes were accurately escribe by the Gaussian-Lorentzian sum functions, S MI (H) from which the peak-to-peak Lorentzian, H L pp, an Gaussian, H G pp, line withs were etermine an line height ratios were correcte for inhomogeneous broaening. 16 Least-squares fits of the spectra were carrie out as etaile previously. 1-3 The Gaussian Figure 3. EPR spectra ue to 0.1 (top trace, at a relative gain of 64) an 93.6 mm (mile trace) PDT in 70% glycerol at 273 K. The fit an spectrum are inistinguishable; however, a efinite resiual is evient. Furthermore V obs (+1) < 0 an V obs (+1) > 0 the reverse of the signs expecte from exchange. The contribution of exchange is negligible, Ω(T) ) ( components, ue to unresolve hyperfine couplings an the effect of fiel moulation yiele H G pp ) ( G. After correcting for fiel moulation, 21 unresolve hyperfine structure contribute G to H G pp. Results Figures 1, 2, an 3 show experimental spectra of 0.1 mm (first trace) an 93.6 mm (secon trace) PDT in 70% glycerol at 343, 293, an 273 K, respectively. Fits of the secon traces to eq 1 are inistinguishable from the spectra on the scales of Figures 1-3. The resiuals, the spectra minus the fits, are shown as the noisy baselines. The thir traces are the ersion components of the fits. The spectrum an resiual in Figure 1 may be compare with spectra of other nitroxies in various solvents at high values of T/η where exchange ominates in Figure 2 of ref 1 an Figures 2, 4, an 5 of ref 2. Spectra simulate from the rigorous theory an their fits to eq 1are given in Figure 1 of ref 1 an Figure 1 of ref 2. In earlier publications, we have laye all of the components of eq 1; here, we omit the absorption components to save space. In Figure 1 an even up to much higher values of ω e /γ leaing to coalescence of the hyperfine lines into a single line, 2 only the absorption an

6 EPR Line Shifts an Line Shape Changes J. Phys. Chem. A, Vol. 113, No. 17, ersion components in eq 1 are neee to escribe the spectra leaing to unetectable ifferences in the spectra an the fits. Thus, the line shapes employe eq 1, Lorentzian-Gaussian sum functions at low values of ω e /γ, an Lorentzian functions at high values, are in excellent agreement with experimental results. In Figures 2 an 3, in contrast to the near-perfect fits in cases where exchange ominates, we observe that fits to spectra exhibiting significant ipolar broaening show efinite resiuals emonstrating that there are epartures of the line shape from Lorentzian. The epartures are not large; in the worst case, better than 97% of the oubly integrate intensity of the spectra is accounte for by the Lorentzian shape. Nevertheless, an uncertainty that is ifficult to estimate is introuce ue to the resiual. The fits to the spectra in Figures 1-3 lea to H G pp ) 0; thus, if the imperfect fits are ue to inhomogeneous ipolar broaening, such broaening is not etecte as being Gaussian. Note that the central lines in Figures 1-3 show ersion components ue to microwave brige imbalances; these are use to correct the ersion amplitues using eq 3. Also, at 273 K, note the curious inversion of the ersion components giving negative values at low fiel an positive at high, the opposite of that expecte from exchange. Note that the previous sentence was written before we receive the input from one of the referees suggesting that negative values coul be expecte from a more accurate theory of ipolar. 20 Fitting low-temperature spectra such as that in Figure 3 with only absorption components (setting V (M I ) ) 0) yiels resiuals that are very similar to those in Figure 3, that is the line shape error is preominantly ue to the absorption. Figure 4 shows representative values of B tot at various temperatures as functions of c in water (Figure 4a) an in 70% glycerol (Figure 4b); the straight lines are linear least-squares fits. In water, the coefficients of correlation, r, are better than at all temperatures, whereas in 70% glycerol the correlation is less satisfactory, eclining to reach a low of r ) at 273 K. The slopes of the total broaening, B tot /c, are given in Figure 5 as functions of T/η. In water, these slopes are linear with T/η; however, they o not extrapolate to the origin. In 70% glycerol, B tot /c shows an upturn as T/η ecreases towar the origin as shown by the inset. In Figures 4, 5, 6, 7, an 11, ata from three spectra taken one after another are plotte separately to show the reproucibility. The appearance of a single atum means that the ifferences are too small to perceive on the scale of the figures. Figure 6 lays values of -A 0 [Vobs / ] # -, correcte using eq 3, as functions of B tot where the superscript obs enotes the observe value, that is that emerging from the fit to eq 1. Plots of +A 0 [Vobs / ] # + (not shown), are very similar to Figure 6; the error bars show one-half the ifference in the two values. In water, shown in Figure 6a, the plots are, as expecte, straight lines of slope approximately equal to unity at higher temperatures, emonstrating that -A 0 [Vobs / ] # - )+A 0 [Vex / ] # + ) B e. In contrast, in 70% glycerol at lower temperatures, as shown in Figure 6b, plots of -A 0 [Vex / ] # - o not become linear until B tot > 1 G. Aitionally, at lower temperatures, -A 0 [Vex # / ] - an +A 0 [V obs / ] # + both become negative, which cannot come from exchange. The straight lines in Figure 6a are linear leastsquares fits constraine to the origin an in Figure 6b are fits to the highest four concentrations. Clearly, the results in Figure 6b o not conform to assumption 3, that ipolar interactions o not contribute to the ersion for values of B tot near the origin. It is not clear from the fits whether (1) ipolar yiels negative values of (A 0 [V obs / ] ( Figure 4. Total line with vs PDT concentration (a) in water, at 268 K, O; 298 K, 0; 323K, ); an 338 K, 4; an (b) in 70% glycerol, at 273 K, O; 298 K, 0; 323K, ); an 338 K, 4. The straight lines are linear least-squares fits constraine to the origin. The slopes are plotte in Figure 5. Coefficients of correlation, r, are larger than at all temperatures in water. In 70% glycerol, r ) (273 K), r > (298 an 323 K), an r ) (338 K). Figure 5. Total broaening constant versus T/η for water, O; an 70% glycerol, 0. The straight line through the water ata is a linear leastsquares fit yieling B tot /c ) (29.3 ( 0.2) + (0.296 ( 0.001) T/η in G/M with η in cp an r ) The ata in 70% glycerol show an upturn at low values of T/η (inset). or (2) the imperfect fits evient in Figures 2 an 3 lea the search in parameter space to nonphysical values. The input receive from one of the referees suggests that (1) might be expecte. Note that the correction term in eq 4 is negligible for the negative values so we rop the superscript #. If possibility (1) hols, then we conclue that the ipolar ersion is not linear in c reaching a plateau at rather low

7 4936 J. Phys. Chem. A, Vol. 113, No. 17, 2009 Bales et al. Figure 7. Dipole-inuce ersion in 70% glycerol versus concentration. Same symbols as Figure 6b. The average value from eq 20 is plotte; the error bars are one-half the ifference. The soli lines are least-squares fits to eq 21 with c* ) 0, 21, 31, an 39 mm from high to low temperature, respectively. The plateau values [V ip / ( )] ) 0, , , an , respectively. where the superscript ip refers to the ipolar contribution be it ue to possibility (1) or (2). The ata in Figure 6b an at all other temperatures show that values of ( [V ip / ] ( achieve constant values away from the origin; thus, for these higher values of B tot, ([V ip / ] ( /B tot ) 0, an Ω(T) ( )([ V ex # /B tot )([ V obs # /B tot (19) allowing us to evaluate Ω(T) ( from eq 12 an B e from eq 8. Therefore, from eq 18 [ V ip )([ V obs # - Ω(T) V ( B tot (20) pp Figure 6. Values of -A 0 (V obs / ) # - vs the total broaening in (a) water at 268 K, O; 303 K, 0; 338K, 4; an in (b) 70% glycerol at 273 K, O; 293 K, 0; 313K, ); an 352 K, 4. The straight lines through the water ata are linear least-squares fits constraine to the origin (r ) 0.999, 303, an 338 K; r ) 0.991, 268 K). Those through the 70% glycerol ata are linear least-squares fits to the highest 4 concentrations (r ) 0.994, 293 K; r ) 0.998, 313 K; an r ) 0.999, 352 K). In water at all temperatures an in 70% glycerol above 323 K, linear fits constraine to the origin using all concentrations yiel the same slopes as those using the highest four not constraine to the origin. Plots of +A 0 (V obs / ) # + (not shown) are similar. The error bars are one-half the ifference in ( A 0 (V obs / ) # (. Were it not for a ipolar contribution to the ersion, the orinate woul be the broaening ue to exchange, B e,eq7. values of B tot. If possibility (2) hols, then the fact that -[V obs / ] + ) [V obs / ] - even when these quantities are negative, is a curious coincience. In either case, we may write [ V obs # )([ V ex # ( [ V ip (18) Figure 7 shows values of [V ip / ], which are average values of ([V ip / ] ( ; the error bars are the stanar eviations of the two values. The ipolar contribution to the observe ersion reaches a plateau above 40 mm. The soli lines in Figure 7 are nonlinear least-squares fits to [ V ip ] [ ) V ip ( )] tanh(c/c*) (21) where the constant [V ip / ( )] is the plateau value at large c an c* is a characteristic concentration at which the plateau is approache. There is no theoretical basis for eq 21; for now, we use it as a guie to the eye in Figure 7. For possible future reference, we plot the values of the constants [V ip / ( )] an c* in the Supporting Information. Guie by the SE, eq 14, reasoning that at equal values of the iffusion coefficient, Ω(T) ought to be the same, we plot values of Ω(T) (the average of Ω(T) ( ) versus η/t in Figure 8a for the two solvents showing that the fractional broaenings by exchange are not accurately preicte by the SE in ifferent solvents. Interestingly, there is enough information in the EPR

8 EPR Line Shifts an Line Shape Changes J. Phys. Chem. A, Vol. 113, No. 17, Figure 9. Exchange broaening constant of PDT versus T/η in water, O an 70% glycerol, 0. The soli line is the SE preiction, eq 15 an the ashe line is a linear least-squares fit to the water ata. Figure 8. Fractional broaening by exchange of PDT in water, O; an in 70% glycerol, 0; versus (a) T/η an (b) the inverse rotational correlation time. The error bars are one-half the ifference in the values of Ω(T) (. In 70% glycerol, exchange an ipolar broaening are equal at T/η ) 20 ( 1 K/cP, which correspons to η ) 15 cp at 300 K. spectrum to pursue the matter further. The rotational correlation time, τ R, was compute from line-height ratios using stanar formulas 22 an correcte for inhomogeneous line broaening. 16 Accoring to the Stokes-Einstein-Debye relation, τ R is -1 proportional to η/t. In complicate solvents, neither D nor τ R is proportional to T/η; however, their ratio is constant. 10,23 In other wors, the epartures from the SE are similar for translation an rotation. This fact motivates the plot in Figure 8b in which values of Ω(T) are plotte versus τ -1 R showing that the transition between the two solvents is continuous. Robinson et al. 10 write a moifie SE employing a viscosity to a power <1 to unify the results. Another possible reason for this iscrepancy might be the effect of steric hinrance on the spin exchange. Because of hyrogen boning, the steric hinrance factor value 7 an thereby the efficiency of spin exchange may be affecte by the solvation of the NO group in water more than in the water-glycerol mixtures. Figure 9 shows values of B e /c as a function of T/η in the two solvents. The soli line is the SE preiction compute from eq 15, whereas the ashe line is a linear least-squares fit to the ata in water yieling K e ) (0.352 ( 0.001)T/η with r ) The SE is reasonably accurate in preicting values of B e /c in 70% glycerol as shown by the inset to Figure 9. The results in the inset are very similar to those from PDT in a series of alkanes: Figure 5 of part 5. 5 In water the ratio [ B e /c]/[ B e /c] SE ) Berner an Kivelson 13 observe a similar result for itert-butyl nitroxie in water where [ B e /c]/[ B e /c] SE ) 0.65 an pointe out that a ifference Figure 10. Dipolar broaening constant versus η/t of PDT water, O an 70% glycerol, 0. The soli line is the SE preiction, eq 16 an the ashe line is a linear least-squares fit of the 70% glycerol ata to a logarithm. At 300 K, soium oecyl sulfate micelles fall at approximately 0.02 cp/k 24 an moel membranes at about 0.21 cp/k. 14 in the hyroynamic iameter of the probe an the exchange interaction istance in water coul rationalize the result. Values of B ip /c compute from eq 10 using the results in Figures 4 an 9 are plotte versus η/t in Figure 10. The straight line is the SE preiction, eq 16. The inset shows the results near the origin where the ata in water are seen to be in reasonable agreement with the SE an the initial slope in 70% glycerol is also in reasonable agreement. The ata in 70% glycerol rather quickly saturate, reaching only about one-half of the preicte static limit of G/M, eq 17. The soli line through the 70% glycerol ata is a linear least-squares fit of the ata to a logarithm yieling B ip /c ) log(η/t) in G/M with η in cp. The transition from low values of η/t to the static limit has been stuie theoretically by Nevzorov an Free 18,19 for a simplifie system in which no hyperfine structure is inclue. Also, their treatment oes not inclue various averaging effects, so their results can only be viewe qualitatively; nevertheless, Figure 6 of ref 19 shows that ipolar broaening ecreases logarithmically with D. Near the static limit, the ipolar broaening becomes inhomogeneous. 19 Values of A obs /A 0 for PDT in 70% glycerol are plotte versus B tot /A 0 in Figure 11a an versus B e /A 0 in Figure 11b where

9 4938 J. Phys. Chem. A, Vol. 113, No. 17, 2009 Bales et al. Figure 11. Observe line spacing of PDT in 70% glycerol versus (a) the total line broaening at 273 K, 3; 292 K, ; 307 K, 4; 313 K, ); 323 K, 0; an 342 K, O; an versus (b) the exchange line broaening employing the same symbols. Data at all other temperatures are plotte in (b) as small plus signs. In (a), the soli lines are quaratic fits to guie the eye at each temperature an in (b) the ashe line is the fit of all ata to eq 13. Eq 13 preicts a common curve vs the exchange broaening away from the origin as is observe. B e is compute from eqs 19 an 8. The soli lines in Figure 11a are quaratic fits to guie the eye; in Figure 11b the ashe line is a fit of all of the ata to eq 13 yieling κ ) ( The ata at temperatures represente in Figure 11a are represente by the same symbols in Figure 11b. All of the rest of the ata are shown by small plus signs. Note in Figure 11a, at lower temperatures, where the ipolar interaction ominates, A obs /A 0 actually increases slightly with B tot /A 0 in contraiction to assumption 4 that ipolar interactions o not contribute to the line shifts. The formation of a common curve in Figure 11b supports the methos to evaluate Ω(T) an B e. Discussion Our goal in this work was to emonstrate a simple means to separate exchange an ipolar base on the EPR line shape, not appealing to any suppose temperature epenence of the interactions. Accoring to the assumptions, arising from oler motional-narrowing theory, Figure 6b woul have been a series of straight lines passing through the origin, an only two samples at high an low concentrations woul be require; however, that simplicity has thus far elue us. Obtaining the ata in Figure 6b requires extensive sample preparation, ata collection, an analysis. One can separate ipolar an exchange, but it is not yet simple. If, inee the theory in ref 20 proves to be an accurate representation of the experimental results, the separation will be complicate, however, with the ae benefit of more information erive from the ipolar. We also carrie out an extensive investigation employing 15 N PDT in 70% glycerol at concentrations up to 85 mm, which we are unable to report because that concentration is not sufficient to reach the linear portions of plots such as Figure 6b. Despite there being some avantages to using 15 N spin probes, they are inferior in these types of stuies for several reasons: (1) for the same concentration, values of B e are smaller by a factor of 3 / 4 because only collisions between ifferent values of M I prouce broaening, (2) exchange ersion which is a function of B e /A 0 is reuce by a factor of 1.4 because the hyperfine coupling constant is larger by that factor, an at least as important (3) one oes not have the reunancy in the exchange ersion at high an low fiels as a measure of the uncertainty or the central line to correct instrumental ersion. Nevertheless, we note that in the testing stages of the theory in ref 20, these probes coul be quite valuable because they woul not present complications inherent in the secon part of assumption 1 or in assumption 5. Note that it is simple to etect the presence of ipolar by (1) preparing plots comparable to those in Figure 6 an looking for slopes less than unity or (2) preparing plots such as those in Figure 11a an looking for curves that o not fall on top of one another. The more emaning task is to quantitatively compute values of Ω(T). The analysis requires nonlinear least-squares fitting of the spectra to eq 1, which gives nearly perfect fits in the limit Ω(T) f 1 but poorer fits as Ω(T) ecreases towar the ipolar limit Ω(T) f 0. These imperfect fits are ue to our incomplete knowlege of the proper EPR shape of a line broaene by ipolar. There is a small, unknown component to the line shape in aition to the preominant Lorentzian shape. Interestingly, Pake oublets appear in the theoretical line shapes; as seen in Figure 1 of ref 18. Further, the lines are broaene inhomogeneously. 18 We suspecte that both of these woul be etecte by the fitting software as Gaussian contributions to the line shape; however, the best fits yiel H G pp ) 0. The iscrepancy in Figures 2 an 3 is small; however, the experimental line is higher in the wings than a Lorentzian, reminiscent of the result in solis; see, for example, p 135 of ref 7. Fortunately, the values of [V ip / ] inuce either by fitting to an improper line shape or by a true negative ipolar contribution reach constant values at about B tot 1 G, allowing us to evaluate Ω(T) from the slopes of ( [V obs / ] # ( observe at higher values of B tot where the epenence is linear. That this approach is vali is supporte by the fact that values of A obs /A 0 form a common curve when plotte against values of B e /A 0 erive from those slopes (Figure 11). Values of B e /c for PDT in 70% glycerol are in remarkable agreement with those preicte from hyroynamic theory using eq 15. We were surprise by this result because it was known that that the rotation of probes in aqueous solutions of glycerol i not follow the SE preiction very well. 10 These ata together with those publishe recently show that B e /c is very similar in hexane, ecane, hexaecane, an now 70% glycerol. So far, only water is ifferent in that [ B e /c]/[ B e /c] SE ) 0.73, although the variation with T/η is nearly linear an passes through the origin. It is known that in many instances the value of B e /c of a nitroxie is lower in water than aprotic solvents. 7 The steric factor, which can be etermine by both the structure of the nitroxie an the solvation of the NO group, can reuce

10 EPR Line Shifts an Line Shape Changes J. Phys. Chem. A, Vol. 113, No. 17, the spin exchange efficiency of nitroxie collisions. It is possible that the iscrepancy we observe in water is ue to the solvation of NO. The results in Figure 8a show that values of Ω(T) at a given value of T/η are solvent epenent in the case of water an 70% glycerol, so one cannot accurately preict its value from a knowlege of the viscosity of the system. Nevertheless, as a guie, using the results in 70% glycerol, ipolar an exchange contribute about equally (Ω(T) ) 0.5) at T/η ) 20 ( 1 K/cP, which correspons to η ) 15 cp at 300 K. Typical micelles present microviscosities, as estimate from rotational correlation times of spin probes, as follows: soium oecyl sulfate an oecyl trimethylammonium bromie, 5-7 cp; 24 tetraalkylammonium oecyl sulfate 6-8 cp; 25 an a sugar-base oecyl sulfate cp. 26 Therefore, exchange is expecte to ominate in some micelles an ipolar in others. Figure 10 shows that the SE, eq 16, overestimates the values of B ip /c in 70% glycerol even at small values of η/t, whereas the preicte static limit, eq 17, is about a factor of 2 larger that the experimental result. The epenence of B ip /c on η/t becomes eciely nonlinear rather near the origin, which explains why efforts to fit line broaening ue to exchange an ipolar using the sum of eqs 15 an 16 have faile in liquis 13 an have le to iscrepancies in moel membranes. 14 The iscrepancies in preictions of eqs 16 an 17, an the experimental results are not unanticipate because the theoretical preictions are upper value estimates as emphasize by Molin et al. 7 because significant averaging of ipolar can occur by various mechanisms. Dipolar begins to be significant even in water at lower temperatures. It has been impossible in the past, measuring only the linewiths to etect the onset of ipolar. Results like those in Figure 5 have misle workers through the years. For example, the marvelous linearity of B tot /c with T/η in water seeme to show that ipolar was negligible; however, the nonzero intercept is revealing. Such a nonzero intercept has been observe an various suggestions have been propose; 27 however, comparing Figure 9, where B e /c oes pass through the origin, with Figure 5 where B tot /c oes not, shows that neglect of even minor ipolar contributions can explain the apparent anomaly. Even in 70% glycerol, we observe a reasonably linear epenence of B tot /c on T/η for T/η > 10 K/cP (inset, Figure 5), but again with a nonzero intercept, signaling a significant ipolar contribution. Consiering Figure 10, is the pessimism of Berner an Kivelson 13 that EPR is not very promising to stuy iffusion still vali? Above about η/t ) 0.1 cp/k, B ip /c becomes rather insensitive to iffusion. A typical iffusion coefficient in a moel membrane is cm 2 /s; see ref 14 an references therein. Using eq 14, employing a probe raius of 3.5 Å 5 for PDT, this correspons to η/t ) 0.21 cp/k, which falls in a region where the ipolar line broaening is rather flat with respect to D, supporting their conclusion. Thus, in viscous liquis, it oes not seem promising to stuy ipolar to estimate iffusion coefficients, nevertheless, one can still separate exchange an ipolar an rely on the well unerstoo exchange. The precision obviously suffers as Ω(T) f 0. Seconary Effects of Exchange. There are two seconary effects of exchange in the case of 14 N nitroxies: As ω e increases, (1) the outer lines broaen slightly faster than the central line an (2) the oubly integrate intensity of the center line increases at the expense of the outer lines (Table 1). 1,2 The ifferential line broaening effect is small but easily measurable as shown in Figure 7 of part 2, 2 where the outer an central Figure 12. Line shifts versus the observe ersion for 46.7 mm PDT in 70% glycerol at K. The abscissa is the mean of values measure from M I )(1 correcte for instrumental ersion; the error bars shown in the inset are the stanar eviations. The soli line in the inset is a linear least-squares fit of the ata to a quaratic. Spinexchange increases towar the right an ipolar towar the left. The lines shift towar one another both above an below the position of no shift near ersion equal to zero. Negative values of the ersion an line shifts towar the center of the spectrum at low temperatures are preictions of the theory in ref 20. linewiths are plotte separately an compare with the rigorous theory. The ifferential intensity is also easily measurable an can be rather substantial. Figure 6 of ref 2 shows a ramatic ifference in the outer an central lines an Figure 7 compares the ifferential intensities to the rigorous theory. In water an in 70% glycerol at higher temperatures, we fin (not shown) these expecte seconary effects. At low values of T/η, all three lines broaen equally an all three intensities are equal, within experimental uncertainty. This confirms assumptions 1 an 5. In principle, ω e, an therefore Ω(T), coul be estimate by these seconary effects. It woul be very useful to fin a theoretically soun aitional line shape component when ipolar is significant. If the search for a better line shape continues, it may be worth keeping in min the possibility of negative ipolar-inuce ersion, even though these results o not prove it. It will be straightforwar to test propose shapes by incluing them into an equation such as eq 1. Unfortunately, as we have learne from fitting exchange broaene lines, it is not straightforwar to iscover such a shape from resiuals such as those in Figures 2 an The nonlinear least-squares fit fins a minimum in parameter space such that the resiual oes not resemble the missing component. 28 Is the Theory of Ref 20 Vali? One of the referees believes that we have observe results preicte by ref 20 in contrast to assumptions 1-5. Ref 20 preicts that when ipolar ominates, negative values of ([V obs / ] # ( arise an increasing B ip causes lines to shift towar one another. Figure 12 where values of A abs /A 0 for 46.7 mm PDT in 70% glycerol at K are plotte as a function of the mean values of ([V obs / ] # (, shows